Ncert Solutions Maths class 12th Maths Class 12th Matrices

34. If $[\begin{matrix} - 1 & 2 & 3 \\ 5 & 7 & 9 \\ - 2 & 1 & 1 \end{matrix}]$ and $[\begin{matrix} - 4 & 1 & - 5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{matrix}]$ , then verify that

(i) (A + B)'= A'+ B' (ii) (A – B) '= A'– B'

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Vishal Baghel | Contributor-Level 10

4 months ago

(i) Given, A = $[\begin{matrix} - 1 & 2 & 3 \\ 5 & 7 & 9 \\ - 2 & 1 & 1 \end{matrix}]$ and B = $[\begin{matrix} - 4 & 1 & 1 \\ 1 & 2 & 3 \\ - 5 & 0 & 1 \end{matrix}]$

Then, A’ = $[\begin{matrix} - 1 & 5 & - 2 \\ 2 & 7 & 1 \\ 3 & 9 & 1 \end{matrix}]$ and B = $[\begin{matrix} - 4 & 1 & 1 \\ 1 & 2 & 3 \\ - 5 & 0 & 1 \end{matrix}]$

A + B = $[\begin{matrix} - 1 & 2 & 3 \\ 5 & 7 & 9 \\ - 2 & 1 & 1 \end{matrix}]$ + $[\begin{matrix} - 4 & 1 & - 5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{matrix}]$ = $[\begin{matrix} - 1 - 4 & 2 + 1 & 3 - 5 \\ 5 + 1 & 7 + 2 & 9 + 0 \\ - 2 + 1 & 1 + 3 & 1 + 1 \end{matrix}]$ = $[\begin{matrix} - 5 & 3 & - 2 \\ 6 & 9 & 9 \\ - 1 & 4 & 2 \end{matrix}]$

LHS. = (A + B)’ = $[\begin{matrix} - 5 & 6 & - 1 \\ 3 & 9 & 4 \\ - 2 & 9 & 2 \end{matrix}]$

RHS = A’+ B’ = $[\begin{matrix} - 1 & 5 & - 2 \\ 1 & 7 & 1 \\ 3 & 9 & 1 \end{matrix}] + [\begin{matrix} - 4 & 1 & 1 \\ 1 & 2 & 3 \\ - 5 & 0 & 1 \end{matrix}] = [\begin{matrix} - 1 - 4 & 5 + 1 & - 2 + 1 \\ 2 + 1 & 7 + 2 & 1 + 3 \\ 3 - 5 & 9 + 0 & 1 + 1 \end{matrix}]$ = $[\begin{matrix} - 5 & 6 & - 1 \\ 3 & 9 & 4 \\ - 2 & 9 & 2 \end{matrix}]$ = L H S

$∴$ (A + B)’ = A’+ B’

(ii) A – B = $[\begin{matrix} - 1 & 2 & 3 \\ 5 & 7 & 9 \\ - 2 & 1 & 1 \end{matrix}] - [\begin{matrix} - 4 & 1 & - 5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{matrix}] = [\begin{matrix} - 1 - (- 4) & 2 - 1 & 3 - (5) \\ 5 - 1 & 7 - 2 & 9 - 0 \\ - 2 - 1 & 1 - 3 & 1 - 1 \end{matrix}] = [\begin{matrix} 3 & 1 & 8 \\ 4 & 5 & 9 \\ - 3 & - 2 & 0 \end{matrix}]$

L.H.S. = (A − B)’ = $[\begin{matrix} 3 & 4 & - 3 \\ 1 & 5 & - 2 \\ 8 & 9 & 0 \end{matrix}]$

R.H.S. = A’ − B’ = $[\begin{matrix} - 1 & 5 & - 2 \\ 2 & 7 & 1 \\ 3 & 9 & 1 \end{matrix}] - [\begin{matrix} - 4 & 1 & 1 \\ 1 & 2 & 3 \\ - 5 & 0 & 1 \end{matrix}] =$ $[\begin{matrix} - 1 - (- 4) & 5 - 1 & - 2 - 1 \\ 2 - 1 & 7 - 2 & 1 - 3 \\ 3 - (- 5) & 9 - 0 & 1 - 1 \end{matrix}] = [\begin{matrix} 3 & 4 & - 3 \\ 1 & 5 & - 2 \\ 8 & 9 & 0 \end{matrix}]$

$∴$ (A − B)’ = A’ − B’

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Let $A = [\begin{matrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{matrix}]$

Given $A = [\begin{matrix} 1 & 0 & 1 \end{matrix}] = [\begin{matrix} 2 & 0 & 2 \end{matrix}]$ ...(1)

$[\begin{matrix} x_{1} + z_{1} & x_{2} + z_{2} & x_{3} + z_{3} \end{matrix}] = [\begin{matrix} 2 & 0 & 2 \end{matrix}]$

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Given $A = [\begin{matrix} - 1 & 0 & 1 \end{matrix}] = [\begin{matrix} - 4 & 0 & 4 \end{matrix}]$

$[\begin{matrix} - x_{1} + z_{1} & - x_{2} + z_{2} & - x_{3} + z_{3} \end{matrix}] = [\begin{matrix} 4 & 0 & 4 \end{matrix}]$

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Let $A = [\begin{matrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{matrix}]$

Given $A = [\begin{matrix} 1 & 0 & 1 \end{matrix}] = [\begin{matrix} 2 & 0 & 2 \end{matrix}]$ ...(1)

$[\begin{matrix} x_{1} + z_{1} & x_{2} + z_{2} & x_{3} + z_{3} \end{matrix}] = [\begin{matrix} 2 & 0 & 2 \end{matrix}]$

∴ x₁ + z₁ = 2 … (2)

x₂ + z₂ = 0 … (3)

x₃ + z₃ = 0 … (4)

Given $A = [\begin{matrix} - 1 & 0 & 1 \end{matrix}] = [\begin{matrix} - 4 & 0 & 4 \end{matrix}]$

$[\begin{matrix} - x_{1} + z_{1} & - x_{2} + z_{2} & - x_{3} + z_{3} \end{matrix}] = [\begin{matrix} 4 & 0 & 4 \end{matrix}]$

⇒ – x₁ + z₁ = −4 … (5)

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Given $A = [\begin{matrix} 0 & 1 & 0 \end{matrix}] = [\begin{matrix} 0 & 2 & 0 \end{matrix}]$

$[\begin{matrix} y_{1} & y_{2} & y_{3} \end{matrix}] = [\begin{matrix} 0 & 2 & 0 \end{matrix}]$

∴ y₁ = 0, y₂ = 2, y₃ = 0

∴ from (2), (3), (4), (5), (6) and (7)

x₁ = 3, x₂ = 0, x₃ = –1

y₁ = 0, y₂ = 2, y₃ = 0

z₁ = –1, z₂ = 0, z₃ = 3

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34. If $[\begin{matrix} - 1 & 2 & 3 \\ 5 & 7 & 9 \\ - 2 & 1 & 1 \end{matrix}]$ and $[\begin{matrix} - 4 & 1 & - 5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{matrix}]$ , then verify that

(i) (A + B)'= A'+ B' (ii) (A – B) '= A'– B'

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34. If [−123579−211] and [−41−5120131] , then verify that (i) (A + B)'= A'+ B' (ii) (A – B) '= A'– B'

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34. If $[\begin{matrix} - 1 & 2 & 3 \\ 5 & 7 & 9 \\ - 2 & 1 & 1 \end{matrix}]$ and $[\begin{matrix} - 4 & 1 & - 5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{matrix}]$ , then verify that

(i) (A + B)'= A'+ B' (ii) (A – B) '= A'– B'